1 Introduction

Previous experience with ray-tracing sun-disk images on off-axis concave mirrors, led us to investigate trajectories within elliptical billiards.

In 2011, we produced the following animation, which uncovered that certain trajectories were perioic orbits, a gallery of which can be found here.

Shortly after, we created the following video to study the geometry of triangular orbits in the ellipse, where we showed the locus of the orbits’ incenters were apparently elliptical.

Then the following fortuitous sequence of events took place (corrections welcome 😄):

  • Sometime after 2011, possibly via Prof. Koiller, this video was viewed by Prof. Serge Tabachnikov (known for his immortal work on the Geometry of Billiards), who shared it w/ Prof. Alexey Glutsyuk.
  • Prof. Glutsyuk shared the problem with Prof. Olga Romaskevich, at the time a Ph.D. student at ENS de Lyon. Prof. Romaskevich, now a postdoc at IRMAR (Rennes, France), proved the ellipticity of the locus using complexification here.
  • A couple of years later, in 2016, Prof. Ronaldo Garcia of IMPA, produced an elegant complement to Prof. Romaskevich’s proof, using only (computational) algebra and differential curve geometry, published here. Prof. Garcia has subsequently proven the ellipticity of loci traced by other triangular orbit centers (baricenter, circumcenter. etc.).
  • More recently, Prof. Glutsyuk’s graduate student Corentin Fierobe has proven, using complexification methods, that the locus of circumcenters is elliptical

To summarize: indeed, the locus of triangular orbits’ incenters is elliptical as are that of many other triangular orbit centers!

2 Simulation

Below we reproduce the original animation this time using the R ecosystem. Without loss of generality, an ellipse is shown with semi-axes of lengths a = 2, b = 1 centered at the origin (the “x”). The ellipse’s foci, at ±sqrt(a²-b²) = ±sqrt(3), are shown as black dots. A base vertex P(t) is chosen on the ellipse at [a*cos(t),b*sin(t)], for some t in [0,2π] (red dot). Via numeric optimization we compute the other two vertices, shown blue, under the constraint they be bisected by their local normals, shown as inward-pointing arrows. For a given t the incenter (shown as a green dot) is computed by intersecting any two normals. Its locus (shown as dashed green ellipse) is obtained by chaining all points under some angular step.

Notice that the two foci are always contained within the orbits, owing to the fact that via Poncelet’s theorem, the orbits’ caustics are confocal with the original ellipse [insert ref].

3 Constant Perimeter

We also verify a fascinating property about triangular orbits in the elliptic billiard: the perimeter of all orbits is constant (though not the area). The graph below shows the perimeter and area numerically calculated for triangular orbits for all t.

On perimeter constancy, via email Prof. Tabachnikov points out that:

the perimeter of a periodic orbit is constant (for any period, starting with three) since these n-gons are extrema of the perimeter function on inscribed n-gons. This function is constant on the curve consisting of its critical points. A somewhat related fact: the centers of mass (with two possible meanings) of a Poncelet polygon move along conics, as proven in our hoax paper with Rich Schwartz.

And Prof. Koiller adds that:

constancy of length comes from Hamiltonian Dynamics. The family of triangles forms an abstract 2-torus in phase space, all trajectories having same velocity and period, hence same length.

4 Addendum

We developed a geometric method which yields closed-form expressions for the extrema in x and y of the incenter locus. Since we know the latter is elliptical, these correspond to the length of the two semi-axes.

Namely, the method comprises the following steps:

  • Compute a “sideways” elementary orbit (red triangle):
    • Start w/ one vertex at the left apex of the billiard L = ( − a, 0).
    • The top right vertex in this orbit lies at P on the boundary s.t. (L − P) is reflected about the normal at P into a vertical line.
    • The third vertex P′ is vertically-symmetric to P.
    • The normals at P and P′ will meet at the x-extremum of the locus (shown red), i.e., at the intersection of the ray shot from P along its normal, and the x axis.
  • Compute an “upright” elementary orbit (blue triangle):
    • Start w/ one vertex at the top apex of the billiard T = (0, 1).
    • The bottom right vertex in this orbit lies at Q on the boundary s.t. (T − Q) is reflected about the normal at Q into a horizontal line.
    • The third vertex Q′ is horizontally-symmetric to Q
    • The normals at Q and Q′ will meet at the y-extremum of the locus (shown blue), i.e., at the intersection of the ray shot from Q along its normal, and the y axis.

5 Code

Simulation code (in R) can be found here

6 Future work

  • display the loci of other triangle centers such as circumcenter, baricenter, etc.
  • draw vertices and incenter ellipse using formulae derived by Ronaldo Garcia

7 References

  1. Fierobe, C., “On the circumcenters of triangular orbits in elliptic billiard”, 2018, link
  2. Garcia, R., “Centers of Inscribed Circles in Triangular Orbits of an Elliptic Billiard”, 2016, link
  3. Schwartz R. and Tabachnikov, S., “Centers of mass of Poncelet polygons, 200 years after”, 2016, link
  4. Glutsyuk, A., “On odd-periodic orbits in complex planar billiards”, 2014, link
  5. Romaskevich, O., “On the Incenters of Triangular Orbits in Elliptic Billiard”, 2013, link
  6. Tabachnikov, S., “Geometry and Billiards”, 1991, link
  7. Birkhoff, G., “Dynamical Systems”, 1927, 1966 edition

© Dan S. Reznik, 2019